Do you have access to edit the DNS records for mywebsite.azurewebsites.net? If so, you could point the DNS records to mydomain.com.
– TreborMar 3 at 14:22

@Trebor it is going to be more complicated than that. In addition to DNS there will be: Azure domain mapping configurations (like virtual hosts), SSL certificates to obtain, and WordPress settings to tweak. Here is a whole article devoted to the topic: greginks.com/cloud/…
– Stephen Ostermiller♦Mar 4 at 2:23

Thank you for the link. It looks less complicated than the one on azure so I’ll give it a try!
– MaisMar 5 at 16:57

When you say mywebsite.example was bought from Microsoft, do you mean that you purchased it through the "Custom Domains" blade of the Web App, or through some other means?
– Zhaph - Ben DuguidMar 6 at 0:05